Matrices on SDNs appear as filters in graph signal processing,
transition matrices in Markov chains, state matrices in dynamic systems, sensing matrices in sampling theory, and in many more applications [1]}, [2]}–[3]}.
In the literature, their
eigenspaces have been used to understand the
communicability between vertices, spectral
clustering for the network and influence of a vertex on the
network [1]}, [5]}, [6]}–[7]}.
In this letter, we consider complex-valued matrices
\({\bf A}=(A(i,j))_{i,j\in V}\)  on the graph \({\mathcal {G}}=(V, E)\) 
with limited geodesic-width \(\omega ({\bf A})\) , which is the smallest nonnegative integer such that \(A(i, j)=0\)  for all \(i, j\in V\)  satisfying \(\rho (i, j)>\omega ({\bf A})\) .
For a matrix with small geodesic-width, we propose a distributed iterative algorithm
to determine eigenvectors associated with its given eigenvalue, see Section . The proposed algorithm is based on the preconditioned gradient descent approach in [3]}
for inverse filtering, and it can be implemented on SDNs with communication range larger than geodesic-width of the matrix.
Moreover, the algorithm has its computational cost and communication expense
for subsystems equipped at every agent of the SDN being independent on the order of the graph \({\mathcal {G}}\) .
In this letter, we also consider finding principal eigenvectors associated with the minimal/maximal eigenvalue of a Hermitian matrix,
and eigenvectors of a polynomial filter of graph shifts, see Sections  and .
